MECH3610 · Advanced Thermofluids
Compressible Flow: Isentropic Flow & Normal Shocks
The fluids half of the course opens with compressible flow (CLO2), where density changes matter and the Mach number governs behaviour. This chapter covers the speed of sound, stagnation (total) properties, the isentropic relations and area-Mach behaviour of converging-diverging nozzles, and the property jumps across a normal shock. This block drives the 20% assignment and part of the 30% open-book final exam; the standard canon is stated plainly here.
What this chapter covers
- 01Mach number Ma = V/a and the speed of sound a = sqrt(gamma*R*T); compressibility matters when Ma >~ 0.3
- 02Stagnation (total) properties and the adiabatic energy equation h + V^2/2 = h0 = const
- 03Isentropic temperature ratio T0/T = 1 + ((gamma - 1)/2)*Ma^2
- 04Isentropic pressure and density ratios as powers of the temperature ratio
- 05Area-Mach relation and choking at the throat (Ma = 1 at A*) in converging-diverging nozzles
- 06Subsonic vs supersonic nozzle branches and back-pressure regimes
- 07Normal-shock relations: downstream Ma2 < 1 and the static property jumps as functions of Ma1
- 08Stagnation-pressure loss and entropy rise across a shock
Isentropic nozzle: static properties and speed at Mach 2
- +1Temperature ratio T0/T = 1 + ((gamma - 1)/2)*Ma^2 = 1 + 0.2*(2.0)^2 = 1.8, so T = 500/1.8 = 277.8 K.
- +1Pressure ratio p0/p = (T0/T)^{gamma/(gamma-1)} = (1.8)^{3.5} = 7.83, so p = 500/7.83 = 63.9 kPa.
- +1Speed of sound a = sqrt(gamma*R*T) = sqrt(1.4*287*277.8) = sqrt(1.116e5) = 334 m/s, and the velocity V = Ma*a = 2.0*334 = 668 m/s.
- +1With Ma = 2.0, far above the 0.3 threshold, density changes are large and the flow is strongly compressible - the incompressible Bernoulli treatment would be badly wrong here.
Key terms
- Mach number (Ma)
- Ma = V/a, flow speed over local speed of sound a = sqrt(gamma*R*T). Ma < 1 subsonic, Ma = 1 sonic, Ma > 1 supersonic; compressibility must be considered when Ma >~ 0.3.
- Stagnation (total) properties
- The state a flow would reach if brought to rest isentropically: total temperature T0 and total pressure p0. The total enthalpy h0 = h + V^2/2 is conserved in adiabatic flow.
- Isentropic relations
- For a perfect gas, T0/T = 1 + ((gamma-1)/2)Ma^2, with p0/p and rho0/rho equal to this ratio raised to gamma/(gamma-1) and 1/(gamma-1) respectively.
- Choking
- The condition where the throat reaches Ma = 1 (area A*) and the mass flow becomes maximal for the given stagnation state; lowering the back pressure further cannot increase the flow.
- Converging-diverging nozzle
- A nozzle whose area first falls to a throat then rises; it can accelerate a flow from subsonic through sonic at the throat to supersonic in the diverging section, subject to the back-pressure regime.
- Normal shock
- A thin, near-discontinuous compression across which a supersonic flow becomes subsonic (Ma2 < 1); static pressure, temperature and density jump up, stagnation pressure drops and entropy rises.
Compressible Flow: Isentropic Flow & Normal Shocks FAQ
When must I treat a flow as compressible?
As a rule of thumb when the Mach number exceeds about 0.3, because below that density changes are under ~5% and the incompressible model is adequate. Above it, use the compressible relations. Assessing whether compressibility matters for a stated case is itself an explicit learning outcome (CLO2).
What stays constant across a normal shock, and what changes?
The flow is adiabatic, so the stagnation temperature T0 (and total enthalpy) is unchanged, and mass, momentum and energy are conserved. But the process is irreversible: static pressure, temperature and density rise, the Mach number drops below 1, entropy increases, and the stagnation pressure falls.
How does a converging-diverging nozzle reach supersonic flow?
A subsonic flow accelerates in the converging section, reaches Ma = 1 only at the throat (choking), then continues accelerating to supersonic speeds in the diverging section — provided the back pressure is low enough for the fully-supersonic branch. Other back pressures give subsonic-throughout flow or a shock in the diverging part.
How is compressible flow assessed in MECH3610?
It drives the 20% assignment (an extended multi-part analysis) and part of the 30% open-book final exam. The equation sheet's fluids relations are the permitted aid, so the marks are for correct model selection and applying the isentropic and shock relations. This guide states the canon and varies numbers; confirm the assignment and exam details on Moodle.
Exam move
Get fluent with the isentropic ratios as a family built on T0/T = 1 + ((gamma-1)/2)Ma^2, and keep gamma and R for air (1.4 and 287 J/kg-K) at your fingertips. Practise the stagnation-to-static conversion in both directions and the area-Mach reasoning for nozzles, and rehearse a normal-shock problem (given Ma1, find Ma2 and the property jumps). Always work in absolute temperatures and check the Ma >~ 0.3 compressibility judgement, which is an explicit outcome. Because these relations live on the equation sheet, drill selecting and applying them under time pressure rather than memorising them. Confirm the assignment and exam format on Moodle.
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